Answer:
No, there is no value of a that makes the statement |a| = -2 true.
Step-by-step explanation:
The absolute value of a number is the distance of that number from zero on the number line. It is always non-negative, so it can't be -2. The absolute value of a number is denoted by two vertical lines on either side of the number like this |a|.
For example, the absolute value of -5 is 5 and the absolute value of 5 is 5. In both cases, the distance from zero is 5.
In the statement |a| = -2, the absolute value of a is -2, which is impossible as the absolute value can't be negative. Therefore, there is no value of a that makes the statement true.
Answer:
no
Step-by-step explanation:
You want to know if there's any value of the variable 'a' that would make its absolute value be -2.
Absolute valueThe absolute value of a number is always positive. There is no such thing as a number whose absolute value is negative, not -2 or any other negative number.
such a value of 'a' Does Not Exist (DNE)
let f be the function given by , where k is a constant. for what values of k, if any, is f strictly decreasing on the interval (-1, 1)?
For any k ∈ (-∞, o), function f is strictly decreasing on the interval (-1, 1).
In mathematics, a function is a relationship or mapping between two sets of elements, known as the domain and the codomain, that assigns each element from the domain to a unique element in the codomain. It describes how inputs from the domain are transformed or related to corresponding outputs in the codomain.
The given function is f = [tex]\frac{kx}{x^2+1}[/tex] ...(1),
Differentiate (1) w.r.t. x,
[tex]\dfrac{d}{dx}f = \dfrac{d}{dx}\frac{kx}{x^2+1}[/tex]
f' = [tex]k\frac{1-x^2}{(x^2 + 1)^2}[/tex]
Since f is decreasing on (-1, 1),
therefore
f' < 0 on (-1, 1),
[tex]k\frac{1-x^2}{(x^2 + 1)^2}[/tex] < 0 on (-1, 1).
As [tex]\frac{1-x^2}{(x^2 + 1)^2}[/tex] > 0 on (-1, 1),
k < 0 for f' < 0.
Therefore, k ∈ (-∞, o) for f to be strictly decreasing on (-1, 1).
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The complete question is as follows:
Let f be the function given by [tex]\frac{kx}{x^2+1}[/tex], where k is a constant. For what values of k, if any, is f strictly decreasing on the interval (-1, 1)?
On the augmented matrix A below, perform all three row operations in the order given. ((a) followed by (b) followed by (c)) and then write the resulting augmented matrix 1 2 -6 | -5
A = 1 3 5 | 2
-3 -10 -2 | -4
(a) R = 1+12 (b) Rg - 3r+r (c) Rs = 4r2 +13
The resulting augmented matrix is:
1 2 -6 | -5
0 1 -8 | 13
0 0 42 | -177
To perform the three-row operations on the augmented matrix A, we proceed as follows:
(a) R = 1 + 12:
Add 12 times row 1 with row 2,
1 2 -6 | -5
0 -14 -32 | 14
-3 -10 -2 | -4
(b) Rg - 3r + r:
Add 3 times row 2 to row 3, and subtract row 3 from row 1
1 2 -6 | -5
0 -14 -32 | 14
0 -16 -38 | -9
(c) Rs = 4r2 + 13:
Multiply row 2 by 4 and add 13 times row 2 to row 3
1 2 -6 | -5
0 1 -8 | 13
0 0 42 | -177
So the resulting augmented matrix is:
1 2 -6 | -5
0 1 -8 | 13
0 0 42 | -177
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Square ABCD is perfectly inscribed in the circle pictured above. If minor arc AD measures 2π, what is the approximate area of the shaded region?
A. 110
B. 72
C. 36
D. 18
E. 9
The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.
The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.
Therefore, Area of the shaded region = πr²/2
Given that minor arc AD measures 2π, the radius of the circle is 2.
Therefore, Area of the shaded region = π(2)²/2 = 4π = 72
The area of the shaded region is half the area of the circle, which is πr²/2. The minor arc AD measures 2π, so the radius of the circle is 2, and the area of the shaded region is 4π, or 72.
The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.
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Help!
Lauren and Shayla have ½ pound of skittles. They want to share the Skittles between themselves and 2 friends.
How many pounds of skittles will each of them receive?
Answer:
0.125 pounds of skittles
Step-by-step explanation:
Lauren and Shayla have a total of ½ pound of skittles and they want to share them with 2 friends, including themselves, so the total number of people is 2+2 = 4
If they want to divide the ½ pound of skittles equally among 4 people, each person will receive ½ pound / 4 = 0.125 pounds of skittles.
Read the following prompt and type your response in the space provided.
Explain how the associative property can be used to solve this problem.
8/9 / 7/5 x 3/16
The associative property shows that the value of 8/9 / 7/5 x 3/16 is 5/24.
How to calculate the value?The associative property states that the way in which we group the numbers or variables in an arithmetic operation does not affect the result. In other words, we can change the grouping of the numbers or variables by using parentheses without changing the outcome of the calculation.
In this problem, we can use the associative property to group the fractions in a way that makes it easier to simplify.
= 8/9 ÷ 7/5 x 3/16
= (8/9) ÷ (7/5) x 3/16
= 8/9 x 5/7 x 3/16
= 8/9 x 15/112
= 120/1008
= 5 / 24
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A storage tank is completely empty. At 12:00 noon, oil is poured into the top of the tank and a valve is opened at the bottom. Oil is poured into the tank at a rate of 2.1 gallons per minute and the valve at the bottom empties at a rate of 1.9 gallons per minute. If the tank does not fill until 12:00 the next day, how many gallons does the tank hold?
Answer: We can start by using the concept of rates.
At 12:00 noon, oil is poured into the top of the tank at a rate of 2.1 gallons per minute and the valve at the bottom empties at a rate of 1.9 gallons per minute.
As the tank is empty at the beginning, the rate at which the oil is poured into the tank is equal to the rate at which the oil is leaving the tank.
2.1 gallons per minute - 1.9 gallons per minute = 0.2 gallons per minute
As the tank does not fill until 12:00 the next day, the difference between the rate of oil that enters the tank and the rate of oil that leaves the tank is equal to zero over the time frame.
0.2 gallons per minute * 1440 minutes = 288 gallons
So, the tank holds 288 gallons.
Step-by-step explanation:
A mass of 50 g stretches a spring 1.568 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 40 cm/s, and if there is no damping, determine the position u of the mass at any time t. Assume g=9.8 m/s^2. Enter an exact answer, u(t)= When does the mass first return to its equilibrium position? Enter an exact answer. Type "pi" to enter π in text entries. Use symbol π in symbolic entries. t = seconds
The position u of the mass at any time t is u(t) = 0.008sin(25t) m, and the mass first return to its equilibrium position when t = π/25 s.
m = 0.25 kg
L = 0.01568 m
u(0) = 0, u′(0) = 0.2 m/s, where u(t) is position of the mass at any time t.
k = mg/L
= (0.25×9.8)/0.01568
= 156.25 N/m
Now we can write the equation of the system: mu′′ + ku = 0
0.25u′′ + 156.25u = 0
u′′ + 625u = 0
The characteristic equation: r² + 625 = 0, r = ±25i .
The general solution is u(t) = C₁sin(25t) + C₂cos(25t)
u(0) = 0 = C₁ × 0 + C₂ × 1 = C₂, C2=0
u′(0) = 0.2 = 25C₁, C₁ = 0.008
So, u(t) = 0.008sin(25t) m.
Equilibrium position: u(t)=0, 0.008sin(25t)=0, sin(25t)=0, 25t=πn, n∈Z
The mass first returns to its equilibrium position when n = 1 or t = π/25.
--The given question is incorrect; the correct question is
"A mass of 250 g stretches a spring 1.568 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 20 cms, and if there is no damping, determine the position u of the mass at any time t.
Enclose arguments of functions in parentheses. For example, sin(2x).
a. Assume g = 9.8 m/s^2. Enter ans exact answer.
u(t) = ______ m
b. When does the mass first return to its equilibrium position?
Enter an exact answer.
t = ______s"--
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an algorithm has n number of steps. which of the following would not be considered a reasonable number of steps?
As per the concept of algorithm, the reasonable number of steps is 2n.
The term algorithm in math is defined as the procedure for solving a mathematical problem in a finite number of steps that frequently involves repetition of an operation.
Here we have given that an algorithm has n number of steps.
Here we need to find the reasonable number of steps.
While we know the definition of algorithms, then her we need to understand the term reasonable time that is defined as the algorithms with a polynomial efficiency or lower are said to run in a reasonable amount of time.
So, the reasonable number of steps is 2n where n refers the number of steps.
Complete Question:
If an algorithm has n number of steps, then in find out the reasonable number of steps?
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In the 30-60-90 triangle below, side s has a length of________ and side q has a length of_______
A. 16√3, 5
B. 8-5, 16
C. 4, 8√3
D. 4√2, 4√2
E. 4, 4 √3
F. 16√3, 16√3
Answer: Option (E)
4, 4√3
===========================================================
Reason:
The hypotenuse is 8. Half of that is 8/2 = 4, which is the value of "s".
This is the short leg of the right triangle. The short leg is opposite the 30 degree angle. Smallest side opposite the smallest angle.
Then use this 30-60-90 triangle template formula
[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\\text{long leg} = 4\sqrt{3}\\\\[/tex]
When the function f(x) is divided by x-1, the quotient is 1x^2+x-8 and the remainder is -2. Find the function f(x) and write the result in standard form.
Answer: When a polynomial is divided by x-1, the quotient is of the form of ax^2 + bx + c and the remainder is of the form of r. The quotient and the remainder are related by the formula:
f(x) = (x-1)(ax^2 + bx + c) + r
Given that the quotient is 1x^2+x-8 and the remainder is -2, we can substitute these into the above equation:
f(x) = (x-1)(1x^2+x-8) + (-2)
f(x) = x^3 - 9x^2 + (1-8)x +(-2)
f(x) = x^3 - 9x^2 - 7x - 2
Therefore the function f(x) is x^3 - 9x^2 - 7x - 2 in standard form.
Step-by-step explanation:
a room 27 feet by 32 feet is to be carpeted all to wall. the width of the carpet is 27 inches. the length, in yards, of the carpet needed for this room is
The length in yards, of the carpet needed for this room is 128 yards.
length of the room = 32 ft
breadth of the room = 27 ft
so we find area now,
area = length X breadth
area = 32 X 27
area = 864ft^2
The width of the carpet in feet is = 2.25ft
so , to find the length of the carpet needed, you must apply the formula for calculate the area of a rectangle and solve for the length, as following,
area = length X width
864 = length X 2.25
length = 864/ 2.25
length = 384ft
length = 128yards
so the length, in yards, of the carpet needed for this room is 128 yards.
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A local restaurant states that its occupancy, the number of people inside the restaurant, can be at most 65.
This includes both people working in the restaurant and the customers. When they open for the lunch rush, there are 8 employees inside the restaurant and customers start to come in at a steady rate of 3 customers per
minute.
Answer: The local restaurant states that its occupancy, the number of people inside the restaurant, can be at most 65. This includes both people working in the restaurant and the customers. When they open for the lunch rush, there are 8 employees inside the restaurant and customers start to come in at a steady rate of 3 customers per minute.
Given this information, we can set up the following inequality:
Occupancy = 8 + 3t (where t is the number of minutes after the restaurant opens) ≤ 65
This inequality represents the maximum occupancy of the restaurant, taking into account the initial number of employees (8) and the rate at which customers are coming in (3 per minute). To find the number of minutes after the restaurant opens until it reaches its maximum occupancy of 65, we can solve for t:
8 + 3t ≤ 65
3t ≤ 57
t ≤ 19
So, the restaurant will reach its maximum occupancy of 65 people after 19 minutes of being open for the lunch rush.
Step-by-step explanation:
You are using glass tiles to make a picture frame for a square photograph with sides 10
inches long. You want the frame to form a uniform border around the photograph. You have
enough tiles to cover 300 square inches. What is the largest possible frame width x?
Using algebraic expressions, The largest possible frame width x is 5.
When addition, subtraction, multiplication, division, and other mathematical operations are performed on variables and constants, the result is a mathematical statement known as an algebraic expression.
Let's say James and Natalie were playing with matchsticks when they had the idea to create number patterns with them.
James made the number 4 with four matchsticks. In order to create a pattern with two 4s, Natalie added three extra matchsticks. They understood that they could keep adding three matchsticks each round to make an additional "four" by doing so. They deduced from this that, generally speaking, in order to create a pattern with n number of 4s, you need 4+ 3(n-1) sticks. We refer to 4+ 3(n-1) as an algebraic expression in this case.
Let the border's width be x, making the overall frame equal to (10 + 2x)(10 + 2x).
area of boundary hence is (10+2x)(10+2x) - 100
but (10+2x)(10+2x) - 100 = 300
(10+2x)² = 400
10+2x = 20
2x = 10
x = 5
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how to solve 5 3/8 + m = 8
Answer:
m=21/8
Step-by-step explanation:
5 /38+m=8
Move the constant to the right.
43/8+m=8
m=8-43/8
m=21/8
Write a multiplication equation to show an equivalent fraction of 15 using fifteenths.
Explain how the two fractions are equivalent.
A multiplication equation to show an equivalent fraction of 15 using fifteenths would gives 10/75
What is a fraction?A fraction consisting of a quotient and remainder is a mixed fraction. we can convert the mixed fraction to improper fraction by first dividing the numerator by denominator and then taking the quotient as whole number and remainder as the numerator of proper fraction keeping the denominator same.
We have to write an equivalent fraction of 15 using fifteenths.
2/15
So let multiply by 5 .
2 / 15 x 5/ 5 = 10/75
Thus, 10/75 is equivalent expression.
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Whats HL? Please help. Thanks
The measure of the length HL will be 18.5 units for the given figure.
What is geometry?One of the earliest areas of mathematics is geometry, along with arithmetic. It is concerned with spatial characteristics like the separation, shape, size, and relative placement of objects.
Given that in the quadrilateral the two sides are Jk = 12 and GF = 25. The value of HL can be calculated by the theorem,
HL = ( 1 / 2 ) x GF x JK
HL = ( 1 / 2 ) x 25 x 12
HL = 18.5
Therefore, the length of section HL will be 18.5 units.
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The length of the rectangle is seven times the width. The perimeter of the rectangle is 312 feet. Find the width and length of the rectangle.
Answer:
The length is 273 and the width is 39
Step-by-step explanation:
Given:
-The perimeter is 312 feet
-The length is seven times the width
Solution:
Let's say the width is x and the length is y.
So our equation is x + y = 312
But, 7x = y
If we substituted that into our equation, it would be:
7x + x = 312,
8x = 312
Divide by 8 on both sides,
x = 39
y = 273
So the length is 273 and the width is 39
All of the following are true statements. According to your Mini-Lesson, which of them is the more formal definition of a function? A Function is a rule where every input has only one output and every input has an output. A Function describes a particular relationship between two quantities: an input quantity and an output quantity. A Function defines a relationship between two Quantities. A Function is a rule that assigns a single, unique output value to each input value. A Function, when graphed, must pass the vertical line test.
A function is a rule that generates one unique output value for every specified input value. Therefore, the fourth statement is true.
In mathematics, the function is something that relates an input to an output. The output obtained will somehow be related to the input. We write the function as f(x). For example, consider f(x) = x². This means that the function f takes the value of x and squares this value. Like when we input x=4. Then the output is f(x)=4²=16.
So, here see the function assigns a single and a unique output value of 16 for the input value of 4. Therefore, option 4 is correct. That is function gives every input value a unique, separate output value.
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Question 2 Multiple Choice Worth 2 points)
(04.05 MC)
There are 200,000 households in Point City. A local computer repair shop takes a random sample of 50 households and finds that the average number of computers per
household from the sample last year that needed to be repaired was 1.15 0.49. Which of the following is an estimate of the total number of computers that needed repairing
last year in Point City?
Between 66 and 164 computers
Between 115 and 49 computers
Between 132,000 and 328,000 computers
Between 230,000 and 98,000 computers
n
The estimate of the total number of computers that needed repairing last year in Point City is given as follows:
Between 132,000 and 328,000 computers.
How to obtain the estimate?The estimate is obtained applying the proportions in the context of this problem.
The confidence interval is given as follows:
1.15 plus/minus 0.49.
Hence the bounds of the interval are given as follows:
Lower bound: 1.15 - 0.49 = 0.66.Upper bound: 1.15 + 0.49 = 1.64.Hence, out of 200,000 households, the lower bound of the estimate is given as follows:
0.66 x 200,000 = 132,000.
The upper bound is of:
1.64 x 200,000 = 328,000.
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There is a stack of eight cards each given a different number from 1 to 8. Suppose we select a card randomly from the stack, replace it, and then randomly select another card. What is the probability that the first card isn’t even number and the second card is less than five
The probability that the first card isn’t even number and the second card is less than five is given as follows:
1/4 = 0.25 = 25%.
How to obtain the probability?A probability is obtained with the division of the number of desired outcomes by the number of total outcomes.
The probabilities for each event in this problem are given as follows:
First card isn't even: 4/8 = 1/2. (1, 3, 5 and 7 are not even).Second card is less than five: 4/8 = 1/2. (1, 2, 3 and 4 are less than 5).The events are independent, hence the probability is obtained with the multiplication of the two probabilities, as follows:
1/2 x 1/2 = 1/4 = 0.25 = 25%.
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An arithmetic sequence is given below. 15, 18, 21, 24, ... Write an explicit formula for the n term a,. th
Answer:
[tex]a_{n}[/tex] = 3n + 12
Step-by-step explanation:
the nth term ( explicit formula ) of an arithmetic sequence is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
here a₁ = 15 and d = a₂ - a₁ = 18 - 15 = 3 , then
[tex]a_{n}[/tex] = 15 + 3(n - 1) = 15 + 3n - 3 = 3n + 12
someone please answer this
determine the values of k such that the given augmented matrix is that of a consistent linear system
a. The value of k from augmented matrix [tex]\left(\begin{array}{ccc}3&- 5&k\\- 9&15&3\end{array}\right)[/tex] is - 1.
b. The value of k from augmented matrix [tex]\left(\begin{array}{ccc}k&1&- 6\\5&- 1&6\end{array}\right)[/tex] is - 5.
The complete question is in the attachment. The augmented matrix is a consistent linear system if
Has at least one solution.The last column in the matrix is not a pivot column.a. The augmented matrix [tex]\left(\begin{array}{ccc}3&- 5&k\\- 9&15&3\end{array}\right)[/tex]
Making the second row become zero R₂ = R₂ + 3R₁
[tex]\left(\begin{array}{ccc}3&- 5&k\\- 9 \:+\: (3 \times 3)&15 \:+\: (3 \times - 5)&3 \:+\: (3 \times k)\end{array}\right)[/tex]
[tex]\left(\begin{array}{ccc}3&- 5&k\\- 9 \:+\: 9&15 \:+\: (- 15)&3 \:+\: 3k\end{array}\right)[/tex]
[tex]\left(\begin{array}{ccc}3&- 5&k\\0&0&3 \:+\: 3k\end{array}\right)[/tex]
From the second row
0x + 0y = 3 + 3k
For the system to be consistent, then
3 + 3k = 0
3k = - 3
k = - 3/3
k = - 1
b. The augmented matrix [tex]\left(\begin{array}{ccc}k&1&- 6\\5&- 1&6\end{array}\right)[/tex]
Making the second row become zero R₂ = R₂ + R₁
[tex]\left(\begin{array}{ccc}k&1&- 6\\5 \:+\: k&- 1 \:+\: 1&6 \:+\: (- 6)\end{array}\right)[/tex]
[tex]\left(\begin{array}{ccc}k&1&- 6\\k \:+\: 5&0&0\end{array}\right)[/tex]
From the second row
(k + 5)x + 0y = 0
For the system to be consistent, then
k + 5 = 0
k = - 5
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The formula to calculate the future value of investment account FV=PV
(1+i)n, where FV is the future value, PV is the Present Value, i is the number of compounds periods. Calculate the future value of a retirement account, if PV = $50,000, i = 10%, and n is 4. ( Hint: change your interest rate to its decimal value before calculating what it ).
Answer: The formula to calculate the future value of an investment account is FV = PV(1+i)^n, where FV is the future value, PV is the present value, i is the interest rate (in decimal form), and n is the number of compound periods.
Given that PV = $50,000, i = 10%, and n = 4, we can plug these values into the formula:
FV = $50,000(1+0.1)^4
By using the formula, FV = $50,000(1.1)^4 = $50,000(1.4641) = $73,205.50
The future value of the retirement account after 4 years is $73,205.50
Step-by-step explanation:
2. Which of these gives the better price for a item priced at $150: a 10/10/10
discount or a 15/15 discount?
Answer:
10/10/10
Step-by-step explanation:
What is the circumference of a circle with a radius of 21 feet? Use 22 over 7 for π.
132 feet
66 feet
5,544 feet
1,386 feet
Answer:
Option A) [tex]132ft[/tex]
Step-by-step explanation:
It is given that,
Radius of a circle = 21ft
We know that π = [tex]\frac{22}{7}[/tex]
So the circumference of a circle = 2πr
Substituting the values,
= [tex]2X\frac{22}{7} X 21[/tex]
So we get,
[tex]= 2X22X3[/tex]
[tex]= 132ft[/tex]
The required circumference of the circle is 132 feet. Option A is correct.
What is the circumference of the circle?To find the circumference of a circle, we use the formula C = πd, where C is the circumference, π is the mathematical constant pi (approximately equal to 3.14), and d is the diameter of the circle.
Here,
The circumference of a circle with a radius of 21 feet can be found using the formula:
C = 2πr
where "C" is the circumference, "r" is the radius, and "π" is the mathematical constant pi, which is approximately equal to 22/7.
Substituting the values given, we have:
C = 2 x (22/7) x 21
C = 132 feet
Therefore, the circumference of the circle is 132 feet. Answer: 132 feet.
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medians and iqrs. for each part, compare distributions (1) and (2) based on their medians and iqrs. you do not need to calculate these statistics; simply state how the medians and iqrs compare. make sure to explain your reasoning.
Distribution (1) and (2) can be compared by looking at their medians and IQRs. If the medians and IQRs are different, the distributions can be assumed to be different.
When comparing two distributions, one of the most useful metrics is their medians and IQRs. The median is the midpoint of the data set, while the IQR is the difference between the first quartile and the third quartile. If the medians and IQRs are different, it suggests that the two distributions are different. For example, if one distribution has a lower median than the other, it suggests that the majority of data points in the lower distribution are lower than the majority of data points in the higher distribution. Similarly, if the IQR of one distribution is greater than the other, it suggests that the data points in the higher distribution are spread further apart than the data points in the lower distribution. Comparing the medians and IQRs of two distributions can help to determine if the distributions are different
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Solve please? I really need help
Could you also make sure I’m right for these?
The area and heights and bases of the figures are: 8.9, 54 in², 35cm²,20.25m², 168 ft², 6cm, 37.8in² and 5mm respectively
How to find the areas?The figures include triangles and trapezoids. The areas are the floor spaces the figures occupy
1) Area of a triangle = 1/2 *b*h
b = 14.3*2 = 28.6
h=8.9
Area = 1/2 *28.6 * 8.9
Area = 127.27 cm²
2) Area of a trapezoid = 1/2(a+b)h
Area = 1/2 (12+6)*6
Area = 1/2*18*6
= 54in²
3) Area of a trapezoid = 1/2(a+b)h
Area = 1/2 (8+6)*5
= (14*5)/2 = 35cm²
4) Area of a triangle = 1/2 *b*h
Area = 1/2 * 9 * 4.5
=20.25m²
5) Area of a trapezoid = 1/2(a+b)h
Area = 1/2 (18+24)*8
= 168ft²
6) Area of a triangle = 1/2 *b*h
38.4 = 1/2*b*12.8
Making b the subject of the relation we have
76.8 = 12.8b
b=6cm
7) Area of a triangle = 1/2 *b*h
Area = 1/2*8.4*9
=37.8in²
8) Area of a trapezoid = 1/2(a+b)h
35=1/2(6+8)*h
70=14h
Making h the subject we have
h= 5 mm
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int: you may want to draw a triangle to express sin(x2) and cos(x2) in terms of t first, then use a half-angle identity! now, express dx in terms of dt:
dx can be expressed in terms of dt as:
dx = 4t2sin(t2)cos(t2)dt + 2t[1 - cos(4t2)]dt + 2t[1 + cos(4t2)]dt
sin(x2) = 2sin(x)cos(x)
cos(x2) = cos2(x) - sin2(x)
Therefore, dx2 = 2sin(x)cos(x)dx + [cos2(x) - sin2(x)]dx
Using the half-angle identity, sin2(x) = 1/2[1 - cos(2x)] and cos2(x) = 1/2[1 + cos(2x)], we can rewrite dx2 as:
dx2 = sin(x)cos(x)dx + 1/2[1 - cos(2x)]dx + 1/2[1 + cos(2x)]dx
Now, we can express dx in terms of dt using the fact that x = t2:
dx = 2tdt
Therefore, dx2 = 2sin(t2)cos(t2)2tdt + 1/2[1 - cos(4t2)]2tdt + 1/2[1 + cos(4t2)]2tdt
Simplifying, we get dx2 = 4t2sin(t2)cos(t2)dt + 2t[1 - cos(4t2)]dt + 2t[1 + cos(4t2)]dt
Therefore, dx can be expressed in terms of dt as:
dx = 4t2sin(t2)cos(t2)dt + 2t[1 - cos(4t2)]dt + 2t[1 + cos(4t2)]dt
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Convert 13/30 to a decimal and a percent.
Answer:
0.43, 43%
Step-by-step explanation: